Tabela Transformada de Laplace Ogata

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Tabelas para a transformada 
de LaplaceA
A
P
Ê
N
D
I
C
E
2 Sr LF DSU V D L LFLDOP DV DULi LV FRPSO [DV DV I o V FRPSO [DV (P
V L D UD] DE ODV SDU V SDUD D UD VIRUPD D /DSODF DV SURSUL D V UD VIRUPD DV
/DSODF 3RU ILP PR V UD RU PDV D UD VIRUPD D /DSODF VD RV IU T P
DV UD VIRUPD DV /DSODF I o V S OVR LPS OVR
Variáveis complexas.�8P P UR FRPSO [R P PD SDU U DO PD SDU LPD L iULD DPEDV
FR V D V 6 D SDU U DO R D LPD L iULD IRU P DULi LV U PRV mR R T V RPL D
Li F 1D UD VIRUPD D /DSODF LOL]D V D R DomR FRPR DULi O FRPSO [D
2 V D
 v ~
R v p D SDU U DO ~ p D SDU LPD L iULD
Funções complexas.�8PD I omR FRPSO [D PD I omR T P PD SDU U DO
PD SDU LPD L iULD R
 
R VmR T D L D V U DLV 2 Py OR p G Gx y+ R DU P R D ODU q
p ± 2 OR p P L R R V L R D L RUiULR D SDU LU R V L R SRVL L R R
L[R U DO 2 FRPSO [R FR D R p ̅ ±
V I o V FRPSO [DV RUPDOP FR UD DV D D iOLV VLV PDV FR URO OL DU
VmR I o V t RFDV VmR UPL D DV L RFDP SDUD D R DORU
8PD I omR FRPSO [D p L D t LF P PD U LmR V R DV DV V DV UL D DV
[LV LU P VVD U LmR UL D D PD I omR D DOt LFD p D D SRU
lim limds
d G s s
G s s G s
s
G
s s0 0D
D
D
D=
+ -
=
" "D D
^
^ ^
h
h h
RPR D Dv D~ DV SR U D ] UR DR OR R P P UR L IL L R LI U V S U
F UVRV ,VVR SR V U PR V UD R PDV mR V Ui SUR D R DT L SRLV V DV UL D DV FDOF OD DV
DR OR R RLV S UF UVRV VS FtILFRV R V D D Dv D D~ IRU P L DLV D UL D D V Ui
D P VPD SDUD T DOT U R UR S UF UVR D Dv D~ SRU D R OD [LV
3DUD P S UF UVR VS FtILFR D Dv R T VL LILFD T R FDPL R p SDUDO OR DR L[R U DO
limds
d G s G j
G G j
Gx y x y
0 �
�
�
�
v v v vD
D
D
D
= + = +
"vD
^ eh o
3DUD R UR FDPL R VS FtILFR D D~ R T VL LILFD T R FDPL R p SDUDO OR DR L[R LPD
L iULR
limds
d G s j
G j j
G
j G
G
j
x y x y
0 �
�
�~ ~ ~ ~D
D
D
D D
= + =- +
"~D
^ fh p
6 VVDV DV UL D DV IRU P L DLV
G j
G G
j Gx y y x
�
�
�
�
�
�
�
�
v v ~ ~
+ = -
R V DV DV FR Lo V D V LU
G G G Gx y y x
�
�
�
�
�
�
�
�
v ~ v ~
= =-
IRU P VD LVI L DV mR D UL D D G G V Ui L RFDP UPL D D (VVDV DV FR L
o V VmR FR FL DV FRPR FR Lo V D F 5L PD 6 VVDV FR Lo V IRU P VD LVI L DV
D I omR V Ui D DOt LFD
RPR [ PSOR DPRV FR VL UDU D V L
G s s= +^ h
( mR
G j j G jGx yv ~ v ~+ = + + = +^ h
R
G G
1
1
1
ex y2 2 2 2v ~
v
v ~
~=
+ +
+ =
+ +
-
^ ^h h
3R V REV U DU T [F R SDUD R SR R ± R V D v ± ~ VD LVID] DV FR
Lo V D F ±5L PD
G G
G G
1
1
1
2 1
x y
y x
2 2 2
2 2
2 2 2
�
�
�
�
�
�
�
�
v ~ v ~
~ v
v ~ v ~
~ v
= =
+ +
- +
=- =
+ +
+
^
^
^
^
h
h
h
h
6
6
@
@
( mR p D DOt LFD P R R R SOD R [F R P ± UL D D G G
[F R P p D D SRU
ds
d G s G j
G G
j G
j s1
1
1
1
x y y x
2 2
�
�
�
�
�
�
�
�
v v ~ ~
v ~
= + = -
=-
+ +
=-
+
^
^ ^
h
h h
1R T D UL D D PD I omR D DOt LFD SR V U RE L D VLPSO VP S OD UL DomR
P U ODomR j 1 VV [ PSOR
ds
d
s s1
1
1
1
2+
=-
+
c
^
m
h
2V SR RV R SOD R RV T DLV D I omR p D DOt LFD VmR FR FL RV FRPR SR RV GL
i L DR SDVVR T RV SR RV R SOD R RV T DLV D I omR mR p D DOt LFD VmR RPL D
RV SR RV L 2V SR RV VL ODU V P T D I omR R V DV UL D DV P DR
L IL L R VmR RPL D RV 2V SR RV VL ODU V RV T DLV p OD VmR F DPD RV
6 U DR L IL L R T D R D ± V D I omR
SDUD 
779Apêndice A – Tabelas para a transformada de Laplace
L U P DORU IL L R mR OR P ± mR ± V Ui F DPD R SROR RU P 6 
R SROR p RPL D R SROR VLPSO V 6 R SROR p F DPD R SROR V D RU P
UF LUD RU P DVVLP SRU LD
3DUD LO V UDU FR VL U D I omR FRPSO [D
G s
s s s s
K s s
1 5 15
2 10
2= + + +
+ +
^
^ ^ ^
^ ^
h
h h h
h h
R P ] URV P ± ± SRORV VLPSO V P ± ± P SROR
SOR SROR P O LSOR RU P P ± 1R T V RU D ] UR P = ∞. Como para
DORU V O D RV
G s
s
KZ^ h
SRVV L P ] UR ULSOR ] UR P O LSOR RU P P = ∞. Se pontos no infinito forem
L FO t RV Ui R P VPR P UR SRORV ] URV (P U V PR P FL FR ] URV 
± ± = ∞, = ∞, = ∞) e cinco polos ( ± ± ± ±
Transformada de Laplace.�9DPRV IL LU
 PD I omR PSR P T SDUD
 PD DULi O FRPSO [D
a P VtPEROR RS UDFLR DO T L LFD T D UD ]D T O D F DL V U UD VIRUPD D
SRU P LR D L UDO /DSODF e dtst
�
-#
 UD VIRUPD D /DSODF
( mR D UD VIRUPD D /DSODF p D D SRU
a� f t F s e dt f t f t e dtst st= = =
� �
- -^ ^ ^ ^h h h h6 6@ @# #
2 SURF VVR L UVR UPL DomR D I omR PSR D SDU LU D UD VIRUPD D /DSODF
p F DPD R G L G F D R DomR LOL]D D SDUD VL i OD p a�±
UD VIRUPD D L UVD /DSODF SR V U RE L D D SDU LU FRP R D [tOLR D V L
L UDO L UVmR
a�± , 0F s f t j F s e ds t2
1 parast
c j
c j
2
r
= =
�
�
-
+
^ ^ ^h h h6 @ #
R F D DEVFLVVD FR U r FLD p PD FR V D U DO p VFRO L D FRP DORU V S ULRU j SDU
U DO R RV RV SR RV VL ODU V VVLP R FDPL R L UDomR p SDUDO OR DR L[R
~ p VORFD R R L[R P DORU F (VV FDPL R L UDomR ILFD j LU L D R RV RV
SR RV VL ODU V
2 FiOF OR D L UDO L UVmR p DSDU P FRPSOLFD R 1D SUi LFD UDUDP L
OL]DPRV VVD L UDO SDUD D RE omR )U T P VDPRV RV Pp R RV [SD VmR
P IUDo V SDUFLDLV D R R Sr LF
V LU DSU V DPRV D 7DE OD T UD] SDU V UD VIRUPD DV /DSODF
I o V FRP P FR UD DV D 7DE OD T UD] SURSUL D V UD VIRUPD DV
/DSODF
780 Engenharia de controle moderno
TABELA A.1
Impulso unitário δ(
UD L iULR
1
2
!n
t
1
n 1
-
-
^ h sn
 
!
s
n
n 1+
±
s a+^ h
±
s a
1
2+^ h
!n t e1
1 n at1
-
- -
^ h s a n+^ h
± 
!
s a
n
n 1+ +^ h
V ~ ~
~
+
FRV ~ ~+
V ~ ~
~
-
FRV ~ ~-
a e
at
- -^ h s s a+^ h
b a e e
at bt
-
-- -^ h s a s b+ +^ ^h h
b a be ae
bt at
-
-- -^ h s a s b
s
+ +^ ^h h
ab a b be ae
at bt
+
-
-- -^ h; E s s a s b+ +^ ^h h
a
e ate1 1 at at2 - -
- -^ h
s s a
1
2+^ h
a
at e1 1 at2 - +
-^ h
s s a
1
2
+^ h
± V ~ s a ~
~
+ +^ h
Pares de 
transformadas 
de Laplace.
(continua)
781Apêndice A – Tabelas para a transformada de Laplace
± FRV ~ s a
s a
~+ +
+
^ h
sene t
1
1 0 1n t n2
2n 1 1
g
~
~ g g
-
-
g~- ^ h
s sn n
n
g~ ~
~
+ +
, /
tg
e t
1
1 1
1
0 1 0 2
sent n2
2
1
2
n
1 1 1 1
g
~ g z
z
g
g
g z r
-
-
- -
=
-
g~-
-
^
^
h
h
s s
s
n ng~ ~+ +
, /
sen
tg
e t1
1
1 1
1
0 1 0 2
t
n2
2
1
2
n
1 1 1 1
g
~ g z
z
g
g
g z r
-
-
- +
=
-
g~-
-
^
^
h
h
s s sn n
n
g~ ~
~
+ +^ h
± FRV ~
~
~
+^ h
~ ± V ~ 2 2 2
3
~
~
+^ h
V ~ ± ~ FRV ~
2
2 2 2
3
~
~
+^ h
sen2
1
~
~
~+^ h
FRV ~
~
~
+
-
^ h
cos cos1
2
2
1
2 1 2 1
2
2
2!
~ ~
~ ~ ~ ~
-
-^ ^h h 2
1
2 2
2
2~ ~+ +^ ^h h
sen cos2
1
~
~ ~ ~+^ h
~+^ h
(continuação)
782 Engenharia de controle moderno
TABELA A.2
a 
a 
a
dt
d f t sF s f != -^ ^ ^h h h; E
a
dt
d f t s F s sf f0 02
2
2 ! != - - P^ ^ ^ ^h h h h; E
a dt
d f t s F s s f
f t
dt
d f t
0
onde
n
n
n n k
k
n k
k
k
k
1
1
1
1
1
!= -
=
-
=
-
-
-
-
^ ^ ^
^ ^
^
^
h h h
h h
h
h
; E /
a f t dt
s
F s
s f t dt
1
n
t 0
= +
!=
^
^
^h
h
h; ;E E# #
a f t dt s
F s
s
f t dt1n n n k
k
n
k
t
1
1 0
H H= +
!
- +
= =
^ ^
^
^ ^h h
h
h h; ;E E/# # # #
a� f t dt s
F st
=^
^
h
h
; E#
( ) ( )limf t dt F s f t dtse existe
s0 0 0
=
"
� �
^ h# #
a –α 
a α)1( α)] = α ) α ≥ 0
a�tf t ds
dF s
=-^
^
h
h6 @
a�t f t
ds
d F s=-^ ^h h6 @
a�t f t
ds
d F sn n n
n
= -^ ^ ^h h h6 @ 
a� limt f t F s ds t f t
1 1se existe
s t 0
=
"
�
^ ^ ^h h h; E #
a�f a aF as=c ^m h; E
a� f t f d F s F s
t
0
1 2x x x- =1 2^ ^ ^ ^h h h h; E#
a�f t g t j F p G s p dp2
1
c j
c j
r
= -
�
�
-
+
^ ^ ^ ^h h h h6 @ #
Propriedades das 
transformadas 
de Laplace.
783Apêndice A – Tabelas para a transformada de Laplace
3RU ILP DSU V DPRV RLV RU PDV IU T P LOL]D RV DP FRP DV UD V
IRUPD DV /DSODF D I omR S OVR D I omR LPS OVR
7 RU PD R DORU L LFLDO lim limf f t sF s0 t s0+ = =" "�+^ ^ ^h h h
7 RU PD R DORU IL DO lim limf f t sF st s 0� = =" "�^ ^ ^h h h
) omR S OVR
f t t
A t t
A t t1 1
0 0
0= - -^ ^ ^h h h
a�f t t s
A
t s
A e st= - -^ h6 @
) omR LPS OVR
,
, ,
lim para
para
g t t
A t t
t t t
0
0 0
t 0 0
0
0
0
1 1
1 1
=
=
"
^ h
a� lim
lim
g t t s
A e
dt
d t s
dt
d A e
s
As A
1
1
t
st
t
st
0 0
0
0
0
0
0
0
0
0
= -
=
-
= =
"
"
-
-
^ ^
^
^
h h
h
h
6 =
6
@ G
@
784 Engenharia de controle moderno